482 ON PERIODIC IRROTATIONAL WAVES [419
C + cos 4^ I
„ f 37a7 25a"£ 15a27 ig s ) + cos5a?|- -2() --—^-f-^-^aS + Sej ................... (13)*
The constant part has no significance for our purpose, and the term in y can be made to vanish by a proper choice of g.
If we use only «; none of the cosines can be made to disappear, and the
value of g is
g^l-tf-M-Tof ............................ (14)
When -we include also /3, we can annul the term in cos 2# by making
,1K.
' ........................... (15)
and with this value of $
„ 5a4 619a«
But unless a is very small, regard to the term in cos 3# suggests a higher value of ft as the more favourable on the whole.
With the further aid of 7 we can annul the terms both in cos 2# and in cos %x. The value of /3 is as before. _ That of 7 is given by
a"/ 139a«\
^n(l+-ir)>
and with this is associated
The inclusion of S and e does not alter the value of g in this order of approximation, but it allows us to annul the terms in cos 4# and cos 5x. The appropriate values are
afi a7
6==
480' and the accompanying value of 7 is given by
while jB remains as in (15).
We now proceed to consider how far these approximations are successful, for which purpose we must choose a value for a. Prof. Burnside took a = -}. With this value the second term of ft in (15) is nearly one-third of the first (Stokes5) term, and the second term of 7 in (20) is actually larger* than the
[* With the alterations specified in the footnotes on p. 481, the terms in (13) involving ct2/3?/, and (a7, a3/3) cos 3ar, become 2y . — a2/3, and cos 3z ( - ^ a7 + ~ a3/3 J. Then the highest terms in
(16), (17), (18), and (20) become respectively - ^ a«, ~ ( + ^ a? \ - ^ «o, and ~ ( + ~ a the second term in (20) being now little more than half the first when a = J. W. P. S.]*" + 2/3e~^ cos 2a?